Loading...

PPT – Risk, Returns, and Risk Aversion PowerPoint presentation | free to download - id: 4babde-MGIyY

The Adobe Flash plugin is needed to view this content

Risk, Returns, and Risk Aversion

- Return and Risk Measures
- Real versus Nominal Rates
- EAR versus APR
- Holding Period Returns
- Excess Return and Risk Premium
- Variance
- Sharpe Ratio
- Risk Aversion and Capital Allocations
- Risk Aversion and Utility Function
- Capital allocation line
- Optimal Allocations

Road map in this and the next lecture

- Risk and return
- Optimal allocation given risk and return tradeoff
- Two-asset allocation
- Efficient frontier
- Multiple asset allocation
- Capital asset pricing models (CAPM)
- Arbitrage pricing theory (APT)
- Fama-French three-factor model

Nominal and Real Rates

- Nominal rate
- Real rate

Example 5.2 Annualized Rates of Return

Formula for EARs and APRs

See page 128-129

Rates of Return Single Period

HPR Holding Period Return P0 Beginning

price P1 Ending price D1 Dividend during

period one

Rates of Return Single Period Example

- Ending Price 48
- Beginning Price 40
- Dividend 2
- HPR (48 - 40 2 )/ (40)

Excess Return

- Risk free rate
- Excess return
- Also known as risk premium

Scenario or Subjective Returns Example

State Prob. of State r in State .1

-.05

.10 2 .2 .05 3 .4 .15 4 .2 .25 5 .1 .35

E(r)

Variance or Dispersion of Returns

Standard deviation variance1/2

Using Our Example

Var

Mean and Variance of Historical Returns

Arithmetic average or rates of return

Geometric Average Returns

TV Terminal Value of the Investment

g geometric average rate of return

Sharpe Ratio

Sharpe Ratio for Portfolios

Measure of risk-return tradeoff Other concepts

Skewness and Kurtosis page 142-143 Check out

the statistics from page 147-151

Page 187

Figure 5.4 The Normal Distribution

Figure 5.4 The Normal Distribution

Normality and Risk Measures

- What if excess returns are not normally

distributed? - Standard deviation is no longer a complete

measure of risk - Sharpe ratio is not a complete measure of

portfolio performance - Need to consider skew and kurtosis

Skew and Kurtosis

- Skew

- Kurtosis

- Equation 5.19

- Equation 5.20

Figure 5.5A Normal and Skewed Distributions

Figure 5.5B Normal and Fat-Tailed Distributions

(mean .1, SD .2)

Value at Risk (VaR)

- A measure of loss most frequently associated with

extreme negative returns - VaR is the quantile of a distribution below which

lies q of the possible values of that

distribution - The 5 VaR , commonly estimated in practice, is

the return at the 5th percentile when returns are

sorted from high to low.

Expected Shortfall (ES)

- Also called conditional tail expectation (CTE)
- More conservative measure of downside risk than

VaR - VaR takes the highest return from the worst cases
- ES takes an average return of the worst cases

Lower Partial Standard Deviation (LPSD)and the

Sortino Ratio

- Issues
- Need to consider negative deviations separately
- Need to consider deviations of returns from the

risk-free rate. - LPSD similar to usual standard deviation, but

uses only negative deviations from rf - Sortino Ratio replaces Sharpe Ratio

Historic Returns on Risky Portfolios

- Returns appear normally distributed
- Returns are lower over the most recent half of

the period (1986-2009) - SD for small stocks became smaller SD for

long-term bonds got bigger

Historic Returns on Risky Portfolios

- Better diversified portfolios have higher Sharpe

Ratios - Negative skew

Figure 5.10 Annually Compounded, 25-Year HPRs

Risk Aversion

- Risk Aversion
- Risk Love
- Risk Neutral
- Utility function

Utility Function

Utility Function U E ( r ) 1/2 A s2 Where U

utility E ( r ) expected return on the asset

or portfolio A coefficient of risk aversion s2

variance of returns

Computing Utility Scores

If A2, then See page 168

Figure 6.2 The Indifference Curve

Page 166, Table 6.3

Allocating Capital Risky Risk Free Assets

- Its possible to split investment funds between

safe and risky assets. - Risk free asset proxy T-bills
- Risky asset stock (or a portfolio)

Example Using Chapter 6.4 Numbers

The total market value of an initial portfolio is

300,000, of which 90,000 is invested in the

Ready Asset money market fund, a risk-free asset.

The remaining 210,000 is invested in risky

securities 113,400 in equity and 96,600 in

long-term bonds. Find the distribution of this

portfolio.

Expected Returns for Combinations

Combinations Without Leverage

?

?

?

Capital Allocation Line with Leverage

- Borrow at the Risk-Free Rate and invest in stock.
- Using 50 Leverage,
- rc (-.5) (.07) (1.5) (.15) .19
- ?c (1.5) (.22) .33

(No Transcript)

Table 6.5 Utility Levels

Optimal Portfolio

- Maximize the mean-variance utility function
- UE(R)-1/2As2
- Based on the expressions for expected return and

s, we have the expression for optimal allocation

y - Example 6.4 (page 175)

Figure 6.6 Utility as a Function of Allocation to

the Risky Asset, y

Figure 6.7 Indifference Curves for U .05 and U

.09 with A 2 and A 4

Optimal Complete Portfolio on Indifference Curves